Given this prior distribution, we compute the MC estimates \(y^{(1)} , \ldots , y^{(E)}\)  at each point in \(\mathcal {X}_{E}\) . From equation (REF ), \(y^{(i)} = g(\textbf {x}^{(i)}) + e^{(i)}\)  where \(e^{(i)}\)  is a zero-mean normally distributed error term with standard deviation \(\omega ^{(i)}\) . We denote by \(\Delta \)  the \(E \times E\)  diagonal matrix where the \(i\) th entry is \([\omega ^{(i)}]^{2}\) . The distribution of \(g(\mathbf {x}_{*})\)  conditional on the observed \(\mathbf {y}\)  can be shown to be normal with mean \(\mathbf {k}_{*}^\top (K + \Delta )^{-1}\mathbf {y}\)  and variance \(k(x_{*}, x_{*})-\mathbf {k}_{*}^\top (K + \Delta )^{-1}\mathbf {k}_{*}\)  [1]}. Thus, given a prior covariance matrix of the form (REF ) and some MC estimates of \(g\)  at the points \(\mathcal {X}_{E}\) , a conditional predictive distribution of \(g(\mathbf {x}_{*})\)  can be found. It is this distribution which will be used in the optimisation algorithm when deciding which solution should next be evaluated.
